Embedding number fields in fields with class number 1

Question

(Apologies if this question isn't quite research-level: a colleague came across it while preparing a non-examinable bonus lecture on class field theory for an undergraduate algebraic number theory course.)

Let $K$ be a number field. Is there always a finite extension $L / K$ such that $L$ has class number 1?

If $K$ has finite class field tower (i.e. the tower of fields $(K_n)_{n \ge 0}$, where $K_0 = K$ and $K_{n+1}$ is the Hilbert class field of $K_n$, eventually terminates) then that solves the problem. But it's a well-known theorem of Golod and Shafarevich that the class field tower of $K$ doesn't terminate if $K$ is an imaginary quadratic field with enough primes ramified.

The textbook my colleague has been using claims that it follows from Golod-Shafarevich that these fields $K$ cannot be embedded in any number field with class number 1, but this implication isn't clear to me. Golod-Shafarevich shows that $K$ has no finite, solvable, everywhere-unramified extension with class number 1, but that's a much weaker statement, isn't it?

Answer 1

See Proposition 1 on p.231 of Cassels and Frohlich for a proof of the claim in the textbook:

The point is that if such an $L$ exists then $K_1L$ is abelian and unramified over $L$ so it is contained in the Hilbert classfield of $L$ which is $L$ itself. By induction, this implies that $K_i \subset L$ for all $i$ so the class field tower of $K$ must be finite.